Chapter 10

What is a zero of a function?

Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. What is a root of a function?

If the remainder is zero when you divide \(P(x)\) by \(x-c\) then what can you say about \(P(c) ?\)

Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. What is the fundamental theorem of algebra?

What are two ways to determine whether \(c\) is a zero of a polynomial?

State the degree of each polynomial equation. Find all of the real and imaginary roots to each equation. State the multiplicity of a root when it is greater than 1. $$x^{5}-4 x^{3}=0$$

Let \(f(x)=x^{4}-1, g(x)=x^{3}-3 x^{2}+5,\) and \(h(x)=4 x^{4}-\) \(3 x^{2}+3 x-1 .\) Find the following function values by using two different methods. See Example \(I\) \(f(1)\)

Determine whether each given value of x is a zero of the given function. See Example 1. $$x=2, \quad P(x)=x-2$$

State the degree of each polynomial equation. Find all of the real and imaginary roots to each equation. State the multiplicity of a root when it is greater than 1. $$x^{6}+9 x^{4}=0$$

Determine whether each given value of x is a zero of the given function. See Example 1. $$x=3, \quad P(x)=x-3$$

State the degree of each polynomial equation. Find all of the real and imaginary roots to each equation. State the multiplicity of a root when it is greater than 1. $$x^{4}+2 x^{3}+x^{2}=0$$

Determine whether each given value of x is a zero of the given function. See Example 1. $$x=7, \quad P(x)=x+7$$

Let \(f(x)=x^{4}-1, g(x)=x^{3}-3 x^{2}+5,\) and \(h(x)=4 x^{4}-\) \(3 x^{2}+3 x-1 .\) Find the following function values by using two different methods. See Example \(I\) $$ f(-2) $$

State the degree of each polynomial equation. Find all of the real and imaginary roots to each equation. State the multiplicity of a root when it is greater than 1. $$x^{5}-4 x^{4}+4 x^{3}=0$$

Let \(f(x)=x^{4}-1, g(x)=x^{3}-3 x^{2}+5,\) and \(h(x)=4 x^{4}-\) \(3 x^{2}+3 x-1 .\) Find the following function values by using two different methods. See Example \(I\) $$ f(5) $$

State the degree of each polynomial equation. Find all of the real and imaginary roots to each equation. State the multiplicity of a root when it is greater than 1. $$x^{4}-6 x^{2}+9=0$$

State the degree of each polynomial equation. Find all of the real and imaginary roots to each equation. State the multiplicity of a root when it is greater than 1. $$x^{4}-8 x^{2}+16=0$$

Determine whether each given value of x is a zero of the given function. See Example 1. $$x=5, \quad P(x)=x^{2}-5 x$$

Let \(f(x)=x^{4}-1, g(x)=x^{3}-3 x^{2}+5,\) and \(h(x)=4 x^{4}-\) \(3 x^{2}+3 x-1 .\) Find the following function values by using two different methods. See Example \(I\) $$ g(-1) $$

Determine whether each given value of x is a zero of the given function. See Example 1. $$x=1, \quad P(x)=x^{3}-x^{2}+x-1$$

Let \(f(x)=x^{4}-1, g(x)=x^{3}-3 x^{2}+5,\) and \(h(x)=4 x^{4}-\) \(3 x^{2}+3 x-1 .\) Find the following function values by using two different methods. See Example \(I\) $$ g(-2) $$

State the degree of each polynomial equation. Find all of the real and imaginary roots to each equation. State the multiplicity of a root when it is greater than 1. $$(2 x+1)^{2}(3 x-5)^{4}=0$$

Let \(f(x)=x^{4}-1, g(x)=x^{3}-3 x^{2}+5,\) and \(h(x)=4 x^{4}-\) \(3 x^{2}+3 x-1 .\) Find the following function values by using two different methods. See Example \(I\) $$ g(2) $$

State the degree of each polynomial equation. Find all of the real and imaginary roots to each equation. State the multiplicity of a root when it is greater than 1. $$x^{4}-2 x^{2}+1=0$$

Let \(f(x)=x^{4}-1, g(x)=x^{3}-3 x^{2}+5,\) and \(h(x)=4 x^{4}-\) \(3 x^{2}+3 x-1 .\) Find the following function values by using two different methods. See Example \(I\) $$ h\left(\frac{1}{2}\right) $$

State the degree of each polynomial equation. Find all of the real and imaginary roots to each equation. State the multiplicity of a root when it is greater than 1. $$4 x^{4}-4 x^{2}+1=0$$

Find a polynomial equation with real coefficients that has the given roots. $$-3,3$$

Find a polynomial equation with real coefficients that has the given roots. $$-4,2$$

Find a polynomial equation with real coefficients that has the given roots. $$-2 i, 2 i$$

Find a polynomial equation with real coefficients that has the given roots. $$-4 i, 4 i$$

Find the \(x\) -intercepts and discuss the behavior of the graph of each polynomial function at its \(x\) -intercepts. See Example 2 . $$f(x)=3 x-2$$

Find a polynomial equation with real coefficients that has the given roots. $$0, i \sqrt{2}$$

Find a polynomial equation with real coefficients that has the given roots. $$-3, i \sqrt{3}$$

Find a polynomial equation with real coefficients that has the given roots. $$i, 1-i$$

Find a polynomial equation with real coefficients that has the given roots. $$2 i,-i$$

Find a polynomial equation with real coefficients that has the given roots. $$\frac{1}{2},-1$$

Find a polynomial equation with real coefficients that has the given roots. $$-1,2,3$$

Find a polynomial equation with real coefficients that has the given roots. $$-2,3,2$$

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$x^{3}+3 x^{2}+5 x+7=0$$

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$2 x^{3}-3 x^{2}+5 x-6=0$$

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$-2 x^{3}-x^{2}+3 x+2=0$$

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$x^{3}+x^{2}-5 x-1=0$$

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$x^{4}-x^{3}+x^{2}-x+1=0$$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{1}{x^{2}}$$

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$x^{4}-1=0$$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{2}{x^{2}-4 x+4}$$

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$x^{4}+x^{2}+1=0$$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{2 x-3}{x^{2}+x-6}$$

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$x^{6}+3 x^{4}+2 x^{2}+6=0$$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{x}{x^{2}+4 x+4}$$